基于重构核的最小二乘配点法求解封闭声腔响应

基于重构核思想,应用无网格配点法构造近似函数,并利用最小二乘方法的原理解决边界问题,离散控制微分方程,建立求解的代数方程.并将此方法应用于封闭声腔响应的求解,即对亥姆霍兹方程进行离散,建立其最小二乘无网格配点格式.该方法的系数矩阵是对称正定的,因而保证了解的稳定性.通过数值算例分别验证了配点均匀分布与随机分布时此方法的...     

对流占优扩散方程的楔形基无网格法

传统的微分方程数值解方法求解对流占优扩散方程时,往往产生数值震荡现象,为了消除数值震荡,本文构建了一种新的数值求解方法――无网格方法进行数值求解。该方法采用配点法并引入一种新的楔形基函数构建了楔形基无网格方法,不需要网格划分,是一种真正的无网格方法,可以避免因为网格划分而影响计算效率。通过对新的楔形基函数的理论分析,证明了本文方法解的存在唯一性。最后,分别通过一维和二维的数值算例,表明该算法计算精度高,可以有效消除对流占优引起的数值震荡,是一种计算对流占优扩散方程数值解的高效方法。     

一种新的无网格方法与有限元耦合法

本文分析了Belytschko和Huerta提出的无网格方法和有限元耦合法各自存在的问题,提出了一种新的无网格方法与有限元耦合法。Belytschko提出的方法的缺点是,无网格方法子域和有限元法子域的界面必须是规则的,交界域内有限元不能随意划分,交界域内无网格方法的节点也不能随意分布。Huerta提出的方法的缺点是对交界域内无网格方法的节点影响域可能无法覆盖交界域。本文提出的无网格方法与有限元耦合法解决了以上两种方法存在的问题,并保留了无网格方法随意配点的优点、交界面可以不规则、提高了无网格子域内的求解精度,从而提高问题的整体求解精度。然后,建立了弹性力学的无网格方法与有限元法的耦合法。最后给出了数值算例。     

薄板屈曲分析的局部Petrov-Galerkin方法

利用薄板控制微分方程的等效积分对称弱形式和对变量(挠度)采用移动最小二乘近似函数进行插值,研究了无网格局部Petrov-Galerkin方法在薄板屈曲问题中的应用。它不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行,并用罚因子法施加本质边界条件。数值算例表明,无网格局部Petrov-Galerkin法不但能够求解弹性静力学问题,而且在求解弹性稳定性问题时仍具有收敛快,稳定性好,精度高的特点。     

用局部Petrov-Galerkin方法求解不可压超弹性材料问题

用一种修正的无网格局部Petrov-Galerkin方法求解了不可压超弹性材料平面应力问题。在建立求解方程过程中,采用径向基函数耦合多项式构造近似函数,并以Heaviside分段函数作为加权函数简化了刚度矩阵的域积分,引入平面应力假设避免了材料不可压引起的数值求解困难。数值算例表明:该文方法求解不可压超弹性材料平面应力问题具有稳定性好、精度高的特点。     

Cartesian网格结合无网格法处理的二维柱体

本文发展了基于四叉树数据结构的网格生成和二维流动的N-S方程数值求解器及动边界问题的Euler方程求解方法。采用压力梯度或者密度梯度的绝对值作为网格自适应的控制参量,同时采用基于最小二乘法的无网格方法处理对于一般Cartesian网格难于处理的物面边界条件。文中采取了绕方柱流动和绕圆柱流动的经典算例对所发展的方法进行了验证。计算的结果验证了所发展的方法在处理绕流流动时的合理性和有效性。采用Naca0012翼型的几种工况验证了所发展的动网格技术在处理无粘流动的合理性和可行性。从而为数值模拟具有复杂几何外形的流动提供了一种网格布局合理、高效,边界处理简单易行的新思路。     

中厚板接触问题的有限元线法分析

本文从动边界变分原理出发推导了中厚板接触问题的有限元线法离散方程体系,并利用新改进的常微分方程求解器Ｃ０Ｌ９０进行求解。数值算例表明,本法精度高,收敛快,无须反复更改网格划分,是一个求解动边界问题的有竞争力的半解析方法。     

流体力学问题基于核重构思想的最小二乘配点法

将基于核重构思想的最小二乘配点法应用于流体力学问题,给出了离散二维不可压缩粘性流体非线性偏微分方程的最小二乘配点格式。为了检验该方法的有效性,以二维Stokes问题——Couette流动为典型算例,分别研究了正压与负压两种工况作用下Couette流动的速度分布。数值模拟结果表明,无论离散点是均匀分布还是随机分布,均给出了较准确的数值结果。     

基于DB小波无网格法:二维弹塑性问题

该文研究了基于Daubechies(DB)小波无网格方法对弹塑性问题中的分析。利用DB小波尺度函数作为基函数近似未知的场函数,不必类似有限元法和传统的无网格法花很大代价去构造所谓的形函数。该文利用新方法建立了增量格式的二维弹塑性问题的求解方案。二维弹塑性问题的数值算例证明了该方法的稳定性和有效性。     

层合板分析的无网格局部Petrov-Galerkin方法

基于Kirchhoff均匀各向异性板控制方程的等效积分弱形式和对挠度函数采用移动最小二乘近似函数进行插值, 进一步研究无网格局部Petrov-Galerkin方法在纤维增强对称层合板弯曲问题中的应用。该方法不需要任何形式的网格划分, 所有的积分都在规则形状的子域及其边界上进行,其问题的本质边界条件采用罚因子法来施加。通过数值算例和与其他方法的结果比较, 表明无网格局部Petrov-Galerkin法求解层合薄板弯曲问题具有解的精度高、收敛性好等一系列优点。      

A point collocation method based on reproducing kernel approximations

A reproducing kernel particle method with built‐in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations. The design and implementation of a Galerkin‐based reproducing kernel particle method, however, faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions. In this paper we present a point collocation method based on reproducing kernel approximations. We show that, in a point collocation approach, the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach. The point collocation method based on reproducing kernel approximations, however, requires the calculation of higher‐order derivatives that would typically not be required in a Galerkin method, A correction function and reproducing conditions that enable consistency of the point collocation method are derived. The point collocation method is shown to be accurate for several one and two‐dimensional problems and the convergence rate of the point collocation method is addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

The reproducing kernel particle method for two-dimensional unsteady heat conduction problems

The numerical solution to two-dimensional unsteady heat conduction problem is obtained using the reproducing kernel particle method (RKPM). A variational method is employed to furnish the discrete equations, and the essential boundary conditions are enforced by the penalty method. Convergence analysis and error estimation are discussed. Compared with the numerical methods based on mesh, the RKPM needs only the scattered nodes instead of meshing the domain of the problem. The effectiveness of the RKPM for two-dimensional unsteady heat conduction problems is examined by two numerical examples.     

Hermite–Cloud: a novel true meshless method

In this paper, a novel true meshless numerical technique – the Hermite–Cloud method, is developed. This method uses the Hermite interpolation theorem for the construction of the interpolation functions, and the point collocation technique for discretization of the partial differential equations. This technique is based on the classical reproducing kernel particle method except that a fixed reproducing kernel approximation is employed instead. As a true meshless technique, the present method constructs the Hermite-type interpolation functions to directly compute the approximate solutions of both the unknown functions and the first-order derivatives. The necessary auxiliary conditions are also constructed to generate a complete set of partial differential equations with mixed Dirichlet and Neumann boundary conditions. The point collocation technique is then used for discretization of the governing partial differential equations. Numerical results show that the computational accuracy of the Hermite–Cloud method at scattered discrete points in the domain is much refined not only for approximate solutions, but also for the first-order derivative of these solutions.     

Semi-Lagrangian reproducing kernel particle method for fragment-impact problems

Fragment-impact problems exhibit excessive material distortion and complex contact conditions that pose considerable challenges in mesh based numerical methods such as the finite element method (FEM). A semi-Lagrangian reproducing kernel particle method (RKPM) is proposed for fragment-impact modeling to alleviate mesh distortion difficulties associated with the Lagrangian FEM and to minimize the convective transport effect in the Eulerian or Arbitrary Lagrangian Eulerian FEM. A stabilized non-conforming nodal integration with boundary correction for the semi-Lagrangian RKPM is also proposed. Under the framework of semi-Lagrangian RKPM, a kernel contact algorithm is introduced to address multi-body contact. Stability analysis shows that temporal stability of the kernel contact algorithm is related to the velocity gradient between two contacting bodies. The performance of the proposed methods is examined by numerical simulation of penetration processes.     

An adaptive isogeometric analysis meshfree collocation method for elasticity and frictional contact problems

A collocation method has been recently developed as a powerful alternative to Galerkin's method in the context of isogeometric analysis, characterized by significantly reduced computational cost, but still guaranteeing higher-order convergence rates. In this work, we propose a novel adaptive isogeometric analysis meshfree collocation (IGAM-C) for the two-dimensional (2D) elasticity and frictional contact problems. The concept of the IGAM-C method is based upon the correspondence between the isogeometric collocation and reproducing kernel meshfree approach, which facilitates the robust mesh adaptivity in isogeometric collocation. The proposed method reconciles collocation at the Greville points via the discretization of the strong form of the equilibrium equations. The adaptive refinement in collocation is guided by the gradient-based error estimator. Moreover, the resolution of the nonlinear equations governing the contact problem is derived from a strong form to avoid the disadvantages of numerical integration. Numerical examples are presented to demonstrate the effectiveness, robustness, and straightforward implementation of the present method for adaptive analysis.     

A gradient reproducing kernel collocation method for boundary value problems

The earlier work in the development of direct strong form collocation methods, such as the reproducing kernel collocation method (RKCM), addressed the domain integration issue in the Galerkin type meshfree method, such as the reproducing kernel particle method, but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of collocation points for optimal convergence. In this work, we intend to address the computational complexity in RKCM while achieving optimal convergence by introducing a gradient reproduction kernel approximation. The proposed gradient RKCM reduces the order of differentiation to the first order for solving second‐order PDEs with strong form collocation. We also show that, different from the typical strong form collocation method where a significantly large number of collocation points than the number of source points is needed for optimal convergence, the same number of collocation points and source points can be used in gradient RKCM. We also show that the same order of convergence rates in the primary unknown and its first‐order derivative is achieved, owing to the imposition of gradient reproducing conditions. The numerical examples are given to verify the analytical prediction. Copyright © 2012 John Wiley & Sons, Ltd.     

A meshfree weak-strong (MWS) form method for the unsteady magnetohydrodynamic (MHD) flow in pipe with arbitrary wall conductivity

In this paper a meshfree weak-strong (MWS) form method is considered to solve the coupled equations in velocity and magnetic field for the unsteady magnetohydrodynamic flow throFor this modified estimaFor this modified estimaFor this modified estimaugh a pipe of rectangular and circular sections having arbitrary conducting walls. Computations have been performed for various Hartman numbers and wall conductivity at different time levels. The MWS method is based on applying a meshfree collocation method in strong form for interior nodes and nodes on the essential boundaries and a meshless local Petrov–Galerkin method in weak form for nodes on the natural boundary of the domain. In this paper, we employ the moving least square reproducing kernel particle approximation to construct the shape functions. The numerical results for sample problems compare very well with steady state solution and other numerical methods.     

Meshfree point collocation method for elasticity and crack problems

A generalized diffuse derivative approximation is combined with a point collocation scheme for solid mechanics problems. The derivatives are obtained from a local approximation so their evaluation is computationally very efficient. This meshfree point collocation method has other advantages: it does not require special treatment for essential boundary condition nor the time‐consuming integration of a weak form. Neither the connectivity of the mesh nor differentiability of the weight function is necessary. The accuracy of the solutions is exceptional and generally exceeds that of element‐free Galerkin method with linear basis. The performance and robustness are demonstrated by several numerical examples, including crack problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Study of radial basis collocation method for wave propagation

A numerical method based on radial basis functions and collocation method is proposed for wave propagation. Standard collocation and weighted boundary collocation approaches yield significant errors in wave problems. Therefore, a new method based on explicit time integration scheme that can correct the inaccuracy in the solutions and the errors accumulated in time integration is developed. This method can be easily applied for low and high dimensional wave problems. The stability conditions are obtained and the relationships between control parameters and stability are evaluated. Requirement of collocation points in numerical dispersion is studied and nondispersion condition is formulated. Eigenvalue analysis is investigated to evaluate the effectiveness of radial basis collocation method for solving wave problems. Eigenvalue study with and without imposing the boundary conditions are compared. The influences of shape parameters and distribution of collocation points and source points are presented. Numerical examples are simulated to examine and validate the proposed method.